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Journal of Mathematical Physics

Continuous Representation Theory Using the Affine Group

Erik W. Aslaksen, John R. Klauder

Published Dec 1, 1969• 89 citations
Publisher

Abstract

We present a continuous representation theory based on the affine group. This theory is applicable to a mechanical system which has one or more of its classical canonical coordinates restricted to a smaller range than − ∞ to ∞. Such systems are especially troublesome in the usual quantization approach since, as is well known from von Neumann's work, the relation [P, Q] = −iI implies that P and Q must have a spectrum from − ∞ to ∞ if they are to be self-adjoint. Consequently, if the spectrum of either P or Q is restricted, at least one of the operators, say Q, is not self-adjoint and does not have a spectral resolution. Thus Q cannot generate a coordinate representation. This leads us to consider a different pair of operators, P and B, both of which are self-adjoint and which obey [P, B] = −iP. The Lie group corresponding to this latter algebra is the affine group, which has two unitarily inequivalent, irreducible representations, one in which the spectrum of P is positive. Using the affine group as our kinematical group, we have developed continuous representations analogous to those Klauder and McKenna developed for the canonical group, and have shown that the former representations have almost all the desirable properties of the latter.

DOI
10.1063/1.1664833

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We present a continuous representation theory based on the affine group. This theory is applicable to a mechanical system which has one or more of its classical canonical coordinates restricted to a smaller range than − ∞ to ∞. Such systems are especially troublesome in the usual quantization approach since, as is well known from von Neumann's work, the relation [P, Q] = −iI implies that P and Q must have a spectrum from − ∞ to ∞ if they are to be self-adjoint. Consequently, if the spectrum of either P or Q is restricted, at least one of the operators, say Q, is not self-adjoint and does not have a spectral resolution. Thus Q cannot generate a coordinate representation. This leads us to consider a different pair of operators, P and B, both of which are self-adjoint and which obey [P, B] = −iP. The Lie group corresponding to this latter algebra is the affine group, which has two unitarily inequivalent, irreducible representations, one in which the spectrum of P is positive. Using the affine group as our kinematical group, we have developed continuous representations analogous to those Klauder and McKenna developed for the canonical group, and have shown that the former representations have almost all the desirable properties of the latter.
In this paper we describe and apply integral quantization, a procedure based on operator-valued measures and the resolution of the identity.We explore its covariance properties in the context of group representation theory.Three applications based on group representations are carried out.The first one concerns the covariant integral quantization based on the spin one-half irreducible representation of SU(2).In this case we show that the quantization of both the quadratic Hopf variables and the Euler angles reproduces the SU(2) Lie algebra.The second example revisits integral quantization based on the Weyl-Heisenberg group.By completing previous results, we show the universality of the canonical commutation relation in such quantizations.In the last example we revisit and enrich the integral quantization based on the affine group of the real line and we give a short account of a relevant application.
Following a recent group theoretical quantization of the symplectic space S={(phi in R mod 2pi, p>0)} in terms of irreducible unitary representations of the group SO(1,2) the present paper proposes an application of those results to the old problem of quantizing modulus and phase in interference phenomena: The self-adjoint Lie algebra generators K_1, K_2 and K_3 of that group correspond to the classical observables p cos(phi), -p sin(phi) and p > 0 the Poisson brackets of which obey that Lie algebra, too. For the irreducible unitary representations of the positive series the modulus operator K_3 has the positive discrete spectrum {n+k, n=0,1,2,...; k > 0}. Self-adjoint operators for cos(phi) and sin(phi) can then be defined as (K_3^{-1}K_1 + K_1 K_3^{-1})/2 and - (K_3^{-1} K_2 + K_2 K_3^{-1})/2 which have the theoretically desired properties for k >0.32. Some matrix elements with respect to number eigenstates and with respect to coherent states are calculated. One conclusion is that group theoretical quantization may be tested by quantum optical experiments.
Developed are the main ideas of the quantized version of affinely-rigid motion. We base on the usual Schrodinger formulation of quantum me- chanics in the configuration manifold. The latter is given, in our case, by the affine group or equivalently by the semi-direct product ofthe linear group GL(n,R) and the space of translations R n , where n equals the di- mension of the physical space. In particular, we discuss the problem of the dynamical invariance of the kinetic energy under the action of affine group, not only under the isometry subgroup. Technically, the treatment is based on the two-polar decomposition of the matrix of the internal con- figuration space and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of nuclear dynamics or even in apparently exotic phenomena in vibrating neutron stars. And, of course, some more prosaic applications in molecular dynamics are possible. It is well known that every dynamical scheme, both mechanical and field- theoretical, may be formulated in two alternative versions: classical and quan- tum. This is always formally possible, although there are situations when one of those versions looks artificial, either classical or quantum. Besides, there are still some non-solved problems concerning the relationship between classical and quantum levels of description. Roughly speaking, they have to do with the mea- surement phenomena and decoherence processes. There is a whole spectrum of views, like, e.g.:
Quantization is defined as the act of assigning an appropriate C*-algebra [Formula: see text] to a given configuration space Q, along with a prescription mapping self-adjoint elements of [Formula: see text] into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here [Formula: see text] is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of [Formula: see text]. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of [Formula: see text].
The observable algebra O of SO_q(3)-symmetric quantum mechanics is generated by the coordinates of momentum and position spaces (which are both isomorphic to the SO_q(3)-covariant real quantum space R_q^3). Their interrelations are determined with the quantum group covariant differential calculus. For a quantum mechanical representation of O on a Hilbert space essential self- adjointness of specified observables and compatibility of the covariance of the observable algebra with the action of the unitary continuous corepresent- ation operator of the compact quantum matrix group SO_q(3) are required. It is shown that each such quantum mechanical representation extends uniquely to a self-adjoint representation of O. All these self-adjoint representations are constructed. As an example an SO_q(3)-invariant Coulomb potential is intro- duced, the corresponding Hamiltonian proved to be essentially self-adjoint and its negative eigenvalues calculated with the help of a q-deformed Lenz-vector.
We discuss the q deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the q-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. The von Neumann theorem in Quantum Mechanics (QM) states that for systems with finite number of degrees of freedom the representations of the canonical commutation relations (ccr), also called the Weyl representations, are each other unitarily equivalent. In the limit of infinite number of degrees of freedom, namely in Quantum Field Theory